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Let <inline-formula> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> be a binary-input memoryless symmetric (BMS) channel with Shannon capacity <inline-formula> <tex-math notation="LaTeX">$I(W)$ </tex-math></inline-formula> and fix any <inline-formula> <tex-math notation="LaTeX">$\alpha &gt; 0$ </tex-math></inline-formula>. We construct, for any sufficiently small <inline-formula> <tex-math notation="LaTeX">$\delta &gt; 0$ </tex-math></inline-formula>, binary linear codes of block length <inline-formula> <tex-math notation="LaTeX">$O(1/\delta ^{2+\alpha })$ </tex-math></inline-formula> and rate <inline-formula> <tex-math notation="LaTeX">$I(W)-\delta $ </tex-math></inline-formula> that enable reliable communication on <inline-formula> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> with quasi-linear time encoding and decoding. Shannon&#x2019;s noisy coding theorem established the <italic>existence</italic> of such codes (without efficient constructions or decoding) with block length <inline-formula> <tex-math notation="LaTeX">$O(1/\delta ^{2})$ </tex-math></inline-formula>. This quadratic dependence on the gap <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> to capacity is known to be best possible. Our result thus yields a constructive version of Shannon&#x2019;s theorem with near-optimal convergence to capacity as a function of the block length. This resolves a central theoretical challenge associated with the attainment of Shannon capacity. Previously such a result was only known for the erasure channel. Our codes are a variant of Ar&#x0131;kan&#x2019;s polar codes based on multiple carefully constructed local kernels, one for each intermediate channel that arises in the decoding. A crucial ingredient in the analysis is a strong converse of the noisy coding theorem when communicating using random linear codes on arbitrary BMS channels. Our converse theorem shows extreme unpredictability of even a single message bit for random coding at rates slightly above capacity.

Arıkan Meets Shannon: Polar Codes With Near-Optimal Convergence to Channel Capacity